关联度分析
python
import numpy as np
import pandas as pd
#关联度分析
#参考序列
Y_0=[170,174,197,216.4,235.8]
#被比较序列
Y_1=[195.4,189.9,187.2,205,222.7]
Y_2=[308,310,295,346,367]
#初始化序列
X_0=np.array(Y_0)/Y_0[0]
X_1=np.array(Y_1)/Y_1[0]
X_2=np.array(Y_2)/Y_2[0]
#计算绝对差序列
deta1=np.abs(X_0-X_1)
deta2=np.abs(X_0-X_2)
#计算deta1，deta2最小值
min1=np.min([np.min(deta1),np.min(deta2)])
max1=np.max([np.max(deta1),np.max(deta2)])
#计算关联系数yita
rio=0.5
yita1 = [(min1 + rio * max1) / (deta1[i] + rio * max1) for i in range(len(deta1))]#计算关联系数yita1
yita2 = [(min1 + rio * max1) / (deta2[i] + rio * max1) for i in range(len(deta2))]#计算关联系数yita2
# #计算关联度
r1=np.mean(yita1)
r2=np.mean(yita2)
# 创建DataFrame
df = pd.DataFrame({
'yita1': yita1,
'yita2': yita2
})
#更改索引
df.index = ['1','2','3','4','5']
print('关联度分析：')
print(df)
#输出关联度
print('关联度：',r1,r2)关联度分析：
yita1 yita2
1 1.000000 1.000000
2 0.705293 0.878928
3 0.381163 0.380878
4 0.355909 0.452620
5 0.333333 0.387478
关联度： 0.5551396262321364 0.6199809489771715
灰色预测GM(1,1)模型
python
from math import exp
#原始序列
X_0=(6,20,40,25,45,35,21,14,18,15.5,17,15)
#累加生成序列
X_1 = np.cumsum(X_0)
#print('累加生成序列:', X_1)
#构造矩阵B和数据向量Y
B = np.zeros((len(X_1)-1,2))
Y = X_0[1:]
for i in range(len(X_1)-1):
B[i][0] = -0.5*(X_1[i]+X_1[i+1])
B[i][1] = 1
#print('矩阵B:', B)
#print('数据向量Y:', Y)
#计算参数a,b
A = np.dot(np.dot(np.linalg.inv(np.dot(B.T,B)),B.T),Y)
a = A[0]
miu = A[1]
#计算预测模型
X_1_predict = np.zeros(len(X_0))
X_1_predict[0] = X_0[0]
for i in range(1, len(X_0)):
X_1_predict[i] = ((X_0[0] - miu / a) * exp(-a * i) + miu / a)
#预测方程式
print('预测方程式：', 'X_1_(k+1) = ', X_0[0]- miu/ a, '*exp(-', a, '*i) +', miu/ a)预测方程式： X_1_(k+1) = -476.05668934240276 *exp(- 0.0746651965600655 *i) + 482.05668934240276
模型检验
python
#残差检验
X_1_predict = np.zeros(len(X_0))
X_1_predict[0] = X_0[0]
for i in range(1,len(X_0)):
X_1_predict[i] = ((X_0[0]-miu/a)*exp(-a*i)+miu/a)
print('预测值:', X_1_predict)
#累减生成序列X_0_predict
X_0_predict = np.zeros(len(X_0))
X_0_predict[0] = X_0[0]
for i in range(1,len(X_0)):
X_0_predict[i] = X_1_predict[i] - X_1_predict[i-1]
#累减生成序列
print('累减生成序列:', X_0_predict)预测值: [ 6. 40.25030314 72.03643912 101.53569452 128.91260092
154.31985252 177.8991578 199.78202993 220.09052023 238.93789892
256.42928694 272.66224218]
累减生成序列: [ 6. 34.25030314 31.78613597 29.4992554 27.3769064 25.4072516
23.57930529 21.88287213 20.30849029 18.8473787 17.49138802 16.23295524]
python
# X_0_predict
#计算绝对误差
error = np.abs(X_0_predict-X_0)
print('绝对误差:', error)绝对误差: [ 0. 14.25030314 8.21386403 4.4992554 17.6230936 9.5927484
2.57930529 7.88287213 2.30849029 3.3473787 0.49138802 1.23295524]
python
#计算相对误差
error_rate = error/X_0
#输出相对误差,保留四位小数，输出百分比
print('相对误差:', np.round(error_rate,4)*100,'%')相对误差: [ 0. 71.25 20.53 18. 39.16 27.41 12.28 56.31 12.82 21.6 2.89 8.22] %
python
#输出检验表，包括原始序列，预测序列，残差序列，绝对误差，相对误差
df = pd.DataFrame({
'原始序列': X_0,
'预测序列': X_1_predict,
'残差序列': X_0_predict,
'绝对误差': error,
'相对误差': error_rate
})
df.index = ['1','2','3','4','5','6','7','8','9','10','11','12']
print('检验表：')
print(df)检验表：
原始序列 预测序列 残差序列 绝对误差 相对误差
1 6.0 6.000000 6.000000 0.000000 0.000000
2 20.0 40.250303 34.250303 14.250303 0.712515
3 40.0 72.036439 31.786136 8.213864 0.205347
4 25.0 101.535695 29.499255 4.499255 0.179970
5 45.0 128.912601 27.376906 17.623094 0.391624
6 35.0 154.319853 25.407252 9.592748 0.274079
7 21.0 177.899158 23.579305 2.579305 0.122824
8 14.0 199.782030 21.882872 7.882872 0.563062
9 18.0 220.090520 20.308490 2.308490 0.128249
10 15.5 238.937899 18.847379 3.347379 0.215960
11 17.0 256.429287 17.491388 0.491388 0.028905
12 15.0 272.662242 16.232955 1.232955 0.082197
相对误差大于于0.5%，模型检验不通过
GM(1,1)残差模型
python
#残差模型进行修正
#对erroe序列进行GM(1,1)模型预测
#原始序列
X_0 = error
#累加生成序列
X_1 = np.cumsum(X_0)
#构造矩阵B和数据向量Y
B = np.zeros((len(X_1)-1,2))
Y = X_0[1:]
for i in range(len(X_1)-1):
B[i][0] = -0.5*(X_1[i]+X_1[i+1])
B[i][1] = 1
#计算参数a,b
A = np.dot(np.dot(np.linalg.inv(np.dot(B.T,B)),B.T),Y)
a = A[0]
miu = A[1]
#计算预测模型
X_1_predict = np.zeros(len(X_0))
X_1_predict[0] = X_0[0]
for i in range(1, len(X_0)):
X_1_predict[i] = ((X_0[0] - miu / a) * exp(-a * i) + miu / a)
# #预测方程式
print('预测方程式：', 'X_1_(k+1) = ', X_0[0]- miu/ a, '*exp(-', a, '*k) +', miu/ a)预测方程式： X_1_(k+1) = -47.4894117559714 *exp(- 0.21716818506240565 *k) + 47.4894117559714
python
#修正后的模型为原始模型加上求导后的方程式
# Differentiate the prediction equation with respect to k
X_1_predict_derivative = np.zeros(len(X_0))
for i in range(1, len(X_0)):
deta3= 1 if (i + 1) >= 2 else 0
X_1_predict_derivative[i] = -a * (X_0[0] - miu / a) * exp(-a * (i-1)) * deta3
# Corrected model by adding the differentiated equation to the original model
X_1_corrected = X_1_predict + X_1_predict_derivative
print('修正后的模型:', X_1_corrected)
#输出修正后的模型方程式
print('修正后的模型方程式：', 'X_1_(k+1) = ', X_0[0]- miu/ a, '*exp(-', a, '*k) +', miu/ a, ' - ', -a*(X_0[0] - miu/a), 'deta(k-1)*exp(-',a, '*k) ')
#输出修正后的模型方程式，分开为k>=2,k<2
print('修正后的模型方程式：', 'X_1_(k+1) = ', X_0[0]- miu/ a, '*exp(-', a, '*k) +', miu/ a, ' - ', -a*(X_0[0] - miu/a), 'deta(k-1)*exp(-',a, '*(k-1)) ', 'k>=2')
print('修正后的模型方程式：', 'X_1_(k+1) = ', X_0[0]- miu/ a, '*exp(-', a, '*k) +', miu/ a, 'k<2')修正后的模型: [ 0. 19.58337881 25.03078704 29.41483178 32.94308733 35.78260836
38.06783956 39.90698129 41.38711264 42.57831436 43.53698707 44.3085217 ]
修正后的模型方程式： X_1_(k+1) = -47.4894117559714 *exp(- 0.21716818506240565 *k) + 47.4894117559714 - 10.31318936072558 deta(k-1)*exp(- 0.21716818506240565 *k)
修正后的模型方程式： X_1_(k+1) = -47.4894117559714 *exp(- 0.21716818506240565 *k) + 47.4894117559714 - 10.31318936072558 deta(k-1)*exp(- 0.21716818506240565 *(k-1)) k>=2
修正后的模型方程式： X_1_(k+1) = -47.4894117559714 *exp(- 0.21716818506240565 *k) + 47.4894117559714 k<2
python
#残差检验
X_1_corrected
#累减生成序列
X_0_corrected = np.zeros(len(X_1_corrected))
X_0_corrected[0] = X_1_corrected[0]
for i in range(1, len(X_1_corrected)):
X_0_corrected[i] = X_1_corrected[i] - X_1_corrected[i-1]
print('累减生成序列:', X_0_corrected)累减生成序列: [ 0. 19.58337881 5.44740823 4.38404474 3.52825555 2.83952103
2.2852312 1.83914174 1.48013134 1.19120172 0.95867271 0.77153463]
python
# X_0_predict
#计算绝对误差
error = np.abs(X_0_corrected-X_0)
print('绝对误差:', error)绝对误差: [0. 6.11131907 4.55574461 1.15568474 7.53661955 1.18744444
0.14100682 2.03499833 0.386519 0.73109566 1.00028958 0.07505627]
python
#计算相对误差
X_0=(6,20,40,25,45,35,21,14,18,15.5,17,15)
error_rate = error/X_0
#输出相对误差,保留四位小数，输出百分比
print('相对误差:', np.round(error_rate,4)*100,'%')相对误差: [ 0. 30.56 11.39 4.62 16.75 3.39 0.67 14.54 2.15 4.72 5.88 0.5 ] %
python
#输出检验表，包括原始序列，预测序列，残差序列，绝对误差，相对误差
df = pd.DataFrame({
'原始序列': X_0,
'预测序列': X_1_corrected,
'残差序列': X_0_corrected,
'绝对误差': error,
'相对误差': error_rate
})
df.index = ['1','2','3','4','5','6','7','8','9','10','11','12']
print('检验表：')
print(df)检验表：
原始序列 预测序列 残差序列 绝对误差 相对误差
1 6.0 0.000000 0.000000 0.000000 0.000000
2 20.0 19.583379 19.583379 6.111319 0.305566
3 40.0 25.030787 5.447408 4.555745 0.113894
4 25.0 29.414832 4.384045 1.155685 0.046227
5 45.0 32.943087 3.528256 7.536620 0.167480
6 35.0 35.782608 2.839521 1.187444 0.033927
7 21.0 38.067840 2.285231 0.141007 0.006715
8 14.0 39.906981 1.839142 2.034998 0.145357
9 18.0 41.387113 1.480131 0.386519 0.021473
10 15.5 42.578314 1.191202 0.731096 0.047167
11 17.0 43.536987 0.958673 1.000290 0.058841
12 15.0 44.308522 0.771535 0.075056 0.005004
修正后的模型检验通过，模型预测准确
预测
{python}
#预测13月份的数据
X_13_predict = ((X_0[0]-miu/a)*exp(-a*12)+miu/a)
print('13月份的预测值:', X_13_predict)13月份的预测值: 76.46124387795606